Whether United Powers Exceed the Same Powers Divided
Forces, sounds, supports, lights, and stars: how many small things joined surpass themselves apart
A text-only folio exploring a single principle across several domains: things united act more powerfully than the same things divided. Leonardo argues that many small forces joined make a greater impetus, that bound rushes support many times the weight each bears alone, and that two columns able to hold a hundred pounds apart will hold three hundred when joined. He extends the idea to sound (many small noises are heard farther together) and to light, noting that many small lights united shine more brightly and that the stars, equal in light to the moon, would form a body far greater than it if joined. A closing note on the pressure of weight explains how an evenly loaded support neither twists nor breaks, but bends and breaks toward the more heavily loaded side.
On this page
Separated powers have less virtue than when united
A heading added near the top edge states the governing principle. Separated powers do not, at one time and for one purpose, make the same virtue and authority as they would if united.
Many small forces united make a greater impetus
Many small forces added together make a greater impetus than the forces do separately. Comparing the impetus of a united force with the parts of a divided force, the united impetus proves the more powerful.
Bound rushes bear twelvefold what each bears alone
One thousand rushes of equal width and length, set upright and separate, each bend under the weight of one dinar. Bound tightly together so they touch, each can support upright twelve times as much weight as before.
Two columns joined bear three hundred pounds
If two columns are each able to support one hundred pounds by themselves when separate, they will support three hundred pounds when joined together.
Many small lights united, and the stars and the moon
Many small luminous bodies united are of greater power than when disunited: lights set along a line and then joined make their place more luminous than before. Likewise the stars, equal in light to the moon, would compose a far greater body if joined—yet our world stays dark when the moon is absent from our hemisphere.
On the pressure of weight: how a support bends and breaks
A support of even thickness and strength, loaded vertically with weight equidistant from its centre, cannot twist or break, though it may be driven in. But an excessive load placed to one side makes it bend toward that side and break in the middle of the opposite side, farthest from the ends.